Can you hear the shape of a manifold? Abstract: Inverse spectral problems ask how much information about an object is encoded in spectral data. For example, Mark Kac's question ``Can you hear the shape of a drum?'' asks whether a plane domain, viewed as a vibrating membrane, is determined by the Dirichlet eigenvalue spectrum of the associated Laplacian, equivalently, by the characteristic frequencies of vibration. This lecture will focus on Kac's question and its generalization to Riemannian manifolds. We will give a partial survey both of positive results, e.g., identifying geometric invariants that are spectrally determined, and of negative results, constructing manifolds with the same spectrum and comparing their geometry.